The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. Since the volume of the cone is given as 12 m³, we can set up the equation as 12 = (1/3)πr²h.
Given that the height of the cone is equal to the diameter of the hemisphere, and the radius of the hemisphere is half the diameter, we can also say that the radius of the cone is equal to the radius of the hemisphere. Let's denote the radius of the hemisphere as r.
Since the radius and the height make a right angle and are perpendicular, we can use the Pythagorean theorem to find the height of the cone. The height of the cone is h, and the length of the hypotenuse is given by √(r² + h²).
Given the information provided, we have:
r + r = 2r is the diameter of the hemisphere = h = 2r
r² + h² = (2r)² = 4r²
h = √(4r² - r²) = √(3r²)
We know that V = 12 = (1/3)πr²h = (1/3)πr² * √(3r²)
12 = √(3) * πr^3 / 3
36 = πr³
r³ = 36 / π
r ≈ 2.6201
Now, the total volume of the object composed of the hemisphere and the cone is the sum of the volume of the cone and hemisphere. The volume of the hemisphere is (2/3)πr³. Substituting r ≈ 2.6201 into both formulas, we get:
V_total = (2/3)π(2.6201)³ + (12 + 2/3)π(2.6201)² * 2.6201
≈ 18 m³
Therefore, the volume of the whole object is approximately 18 m³. So, the correct answer is:
18 m³
Use the image to answer the question.
A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m3 . What is the volume of the whole object?(1 point)
Responses
24 m3
24 m cubed
36 m3
36 m cubed
There is not enough information to say.
There is not enough information to say.
18 m3
1 answer
4 answers